Math Heuristics

Math Heuristics

Introduction

What is a Mathematics Learning Disability?

It is estimated that between 5 and 8% of students have a learning disability (LD) in mathematics (Geary, 2004). The definition of what constitutes a math LD is vague and narrow, with most research offering an exclusionary definition – what it is definitely not – as opposed to typical definitions of specific word-level reading disabilities, such as problems with word recognition or reading fluency (Fletcher, Lyon, Fuchs, & Barnes, 2007).

Thus, a student is considered to have a math LD if they struggle in their “mathematics abilities,” but have “average- or above average-ability IQ [intelligence quotient], normal sensory function, adequate educational opportunity, and [are absent] of any other developmental disorders and emotional disturbances” (Fletcher et al., 2007, p. 207). Math LDs can be specific, such as dyscalculia, or more general, as Wright (2011) points out:

One type of learning disability affecting mathematics can stem from an individual’s difficulty processing language, another might be related to visual spatial confusion, while yet another could include trouble retaining math facts and keeping procedures in the proper order. While extremely rare, there are some learners who cannot successfully compare the lengths of two sticks and others who have almost no ability to estimate. Finally, some people experience emotional blocks so overwhelming as to preclude their ability to think responsibly and clearly when attempting math, and these students are disabled, as well. (Wright, 2011)

Although research has distinguished between math LDs and LDs in reading, the use of interventions in the math classroom, such as heuristics (described later), are not limited to just students with math LDs. A number of researchers have found that many students with LDs, including those with specific reading and writing LDs, have difficulties in math as well (e.g. Fuchs & Fuchs, 2003; Geary, 1993, 2004; Hutchinson, 1993).

Choosing appropriate interventions

The techniques described in this summary are also useful for students who experience early difficulties in math not related to learning disabilities, as math “challenges may persist well into the upper elementary grades” (e.g. Jordan, Kaplan, Ramineni, & Locuniak, 2009; as cited in Mastropieri, Scruggs, Hauth, & Allen-Bronaugh, 2012, p. 221).

The most important consideration when choosing a heuristic or other intervention for math LDs, is to pick an intervention that matches the problems the student is experiencing.

In a study conducted by Burns (2011), students were assigned to receive either a procedural intervention (enhancing the proficiency of steps, without re-teaching the actual concept) or conceptual intervention (re-teaching the concept in a different way). There was significant improvement found when the intervention matched the difficulties that student was experiencing - a procedural intervention for a student who struggles with the concept is less effective than a conceptual intervention. Therefore, “assessment is critically important for math intervention” (Burns, 2011, p. 215; see also Chard, Ketterlin-Geller, & Jitendra, 2008).

Heuristics are best used with students who understand the mathematic concept, but who have trouble remembering the steps in completing a problem (Burns, 2011; Snyder, 1988).

What is a heuristic?

At its most basic, a heuristic is “a short cut in problem solving; it is a rule for reducing the number of mental operations (or information-processing steps) taken to solve a problem” (Gray, 1994, p. 395).

Sometimes heuristics are taught by teachers, sometimes students come across them on their own. It is important to note that heuristics are general strategies that a student can use on their own to help identify and solve a math problem (Gersten et al., 2009).

Peer tutoring, a simplification of the problem by the teacher, or the use of math implements (such as a calculator or ruler) are not considered heuristics. Neither are algorithms – for example, the slope intercept equation (y = mx+b) is not considered a heuristic (Siew, Hedberg, & Lioe, 2005).

A general heuristic might be something like (Gersten et al., 2009):

Read the problem.

Highlight the key words.

Solve the problems.

Check your work.

Other examples might include using a rule of thumb, an educated guess, an intuitive judgement, stereotyping, or common sense. The most basic heuristic is considered to be trial and error (“Heuristics”, 2014).

Math teachers use a number of heuristics in their class already. The acronym FOIL (first, outside, inside, last), used when expanding algebraic expressions, is a widely used heuristic. A similar acronym, BEDMAS (brackets, exponents, division, multiplication, addition, subtraction), used when solving an equation with multiple operations, is also widely used.

Heuristics Use in the Math Class

Key Word Strategy

The most basic strategy suggested for students with learning disabilities is what Kelly and Carnine (1996) call the “key-word” strategy. Although using the strategy can be overly simplistic and cause problems, many students are taught it as a base heuristic, or, more often, they learn it on their own.

Orosco (2014) taught this strategy to English language learners (ELL) and poor readers who struggled in math, finding it a successful base strategy. He updated the name, calling it dynamic strategic math (DSM), or vocabulary modification.

The “key-word” strategy involves associating common words with the operation they represent. For example, they might associate “gave away” to mean the question involves subtraction.

For example...

In the word problem “Amy had 10 pencils. She gave away 4 to Meaghan. How many pencils does Amy have left?” this strategy would work. However, “gave away” takes on a different meaning in the problem “Amy gave away 4 pencils to Meaghan. She gave away 6 pencils to Mino. How many pencils did Amy give away?” and the problem does not involve subtraction.

As Kelly and Carnine (1996) point out, while this heuristic is helpful for students with math LDs, it is actually “too narrow to facilitate the correct solution of many word problems” (p. 1), and should be taught to students as a last resort if no other strategy works.

Underlining Important Information

Landi (2001) suggests a number of strategies students could use to simplify a problem. One common characteristic of word problems is that they tend to have extraneous information. Students should underline or highlight the important information, allowing for a simplification of the process.

For example...

Monkeys like to live in trees and eat bananas. If a monkey picks three bananas in the morning and three in the afternoon, and then his friend gives him five bananas, how many bananas does the monkey have to eat?

The process of teaching this strategy is as important as its eventual use. Landi (2001) suggests doing several word problems together, underlining the important information, and discussing why certain parts were highlighted and certain parts were not.

These two are not the only examples – Miller and Strawser (1996) discuss numerous other mnemonic devices researchers have developed to help students with learning disabilities succeed in math.

Researchers Miller and Mercer (1993) have devised two mnemonic devices to “cue students to answer a math fact from memory, if possible, and to use previously learned strategies to determine the answers of any problems not yet committed to memory” (p. 79). Those two strategies are known as DRAW and SOLVE. Both of these techniques are used for simple math facts, such as 6 x 8.

Miller and Mercer have further developed DRAW to help with the solving of word problems. Students can use FAST DRAW to remember the steps. This is particularly effective as students can build upon an existing strategy when they move from math facts to more complex word problems.

SIGNS, developed by Watanabe (1991) as part of his doctoral thesis, treads similar ground to both SOLVE and DRAW, but using different steps. If students are having trouble with using SOLVE or DRAW, SIGNS may provide enough extra cues to solve the problem.

Heuristics for problem-solving

One of the earliest studies in the use of heuristics for students with LDs solving word problems was done by Hutchinson (1993).

Twenty adolescents with LDs between the ages of 12 and 15, who were a minimum of three years behind typical scores on a standard achievement test in math, were divided into a control and instruction group. Students in the instruction group received instruction on “problem solving individually in a learning disabilities resource setting during regular school hours” (Hutchinson, 1993, p. 40).

Strategies involved questions students asked themselves at two stages:

When representing the problem (turning the word problem into an equation), and

As with most strategies, Hutchinson (1993) modelled the use of these strategies to students beforehand. The results showed that “strategy instruction is an effective approach” (Hutchinson, 1993, p. 49). Students receiving strategy instruction showed improvement in overall scores, with most students increasing from a score of 0/5 on questions pre-intervention, to 4/5 or higher post-intervention.

Coping with Math Anxiety

Past research has suggested the strong role students’ attitudes and beliefs play in learning and succeeding in mathematics (Mercer & Miller, 1992; National Council of Supervisors of Mathematics, 1988; National Council of Teachers of Mathematics, 1989). One of the main areas of student negativity towards math may come from mathematics anxiety, a common problem when instructing students with learning disabilities.

Kamann and Wong (1993) created a number of coping self-statements for students with LDs who experience math anxiety, although the statements could be effective for any student experiencing math anxiety, regardless of exceptionality. Twenty students, 10 of whom were diagnosed with LDs, were given an initial baseline test to see how they coped with frustrating and difficult math problems. Investigators then modelled the use of the coping strategy while solving a similar problem. Two oversized cue cards were used to remind students of the steps.

The first contained the coping process:

assessment of the situation,

recognizing and controlling the impulse of negative thoughts; and

reinforcing.

This card was supplemented with a second card outlining sample phrases students could use at each step:

Assessment of the situation

What is it I have to do?

Look over the task and think about it

Recognizing and controlling the impulse of negative thoughts

Recognition:

Okay, I feel worried and scared …

I’m saying things that don’t help me …

I can stop and think more helpful thoughts.

Confronting/Coping/Controlling:

Don’t worry. Remember to use your plan.

Take it step by step—look at one question at a time.

Don’t let your eyes wander to other questions.

Don’t think about what others are going. Take it one step at a time.

When you feel your fears coming on … take a deep breath, think “I am doing just fine. Things are going well.”

Reinforcing

I did really well in not letting this get the best of me.

Good for me, I did a good job.

I did a good job in not allowing myself to worry so much.

Students were then asked to complete similar questions to their baseline test, while verbalizing their use of the coping process. Students with LDs did significantly better when using the coping statements, with the number of correct questions jumping from an average of 23% in the pre-test to 57.9% in the post test. Similar results were found on a second post-test. It appears that teaching students to use positive self-coping strategies while completing math questions is a positive strategy, leading to increased math scores.

Conclusion

The use of heuristics, or self-strategies, in mathematics can have a profound impact on a student’s ability to quickly and accurately solve a math fact or word problem. Students with LDs (whether math specific or not) will especially benefit from the structure and sequence a heuristic provides.

The most important thing to note about the use of heuristics is the need for modelling of their use. All of the studies mentioned within this review are consistent in that they highlight a clear need for the teacher to use and model the heuristic.

These guides, written by Mercer and Miller, are filled with student-centered heuristic strategies for learning basic math facts. The series can be ordered at http://www.edgeenterprisesinc.com/

This article, written by a member of the National Center for Learning Disabilities, provides an easy explanation about what constitutes a learning disability in mathematics, as well as how to assess a student who may have a mathematics disability. It also has a brief discussion on simple techniques that could prove useful when helping a student who is having difficulties in math.

Watanabe, A. (1991). The effects of a mathematical word problem solving strategy on problemsolving performance by middle school students with mild disabilities. Unpublished doctoral dissertation, University of Florida, Gainesville.

Kyle Robinson is entering his second year in the Master of Education program at Queen’s University, with a focus on the Inclusion of Exceptional Students. Kyle is an OCT certified teacher (I/S), and has taught in schools in the Limestone and Toronto District School Boards. Besides inclusion, Kyle’s research interests also include the Psychology of Learning Disabilities, Special Education programs in Secondary Schools, and the History and Philosophy of Education.

Nancy L. Hutchinson is a professor of Cognitive Studies in the Faculty of Education at Queen’s University. Her research has focused on teaching students with learning disabilities (e.g., math and career development) and on enhancing workplace learning and co-operative education for students with disabilities and those at risk of dropping out of school. In the past five years, in addition to her research on transition out of school, Nancy has worked with a collaborative research group involving researchers from Ontario, Quebec, and Nova Scotia on transition into school of children with severe disabilities. She teaches courses on inclusive education in the preservice teacher education program as well as doctoral seminars on social cognition and master’s courses on topics including learning disabilities, inclusion, and qualitative research. She has published six editions of a textbook on teaching students with disabilities in the regular classroom and two editions of a companion casebook.